Retardation, r, increases
linearly with both the thickness, t, of a specimen and with the birefringence,
n2-n1: the greater the thickness, the greater the
retardation (the thicker the crystal, the farther behind the slow ray
gets from the fast ray); the greater the difference between the refractive
indices (birefringence) to begin with, the greater the retardation (or
higher the interference color). That is,

r
= t(n2-n1)

Or, let’s shorten
it by putting in “B” for the birefringence (n2-n1),

r
= tB

where r is the retardation
expressed in nanometers, nm; t is the thickness, which we measure with
our eyepiece micrometer, and express in micrometers, µm; and B is the
birefringence (the numerical difference between the principal refractive
indices), which is unitless. It will be seen immediately that, in order
to achieve equality of units, the thickness, t, must be multiplied by
1000 (1000 nm/µm) in order to express retardation in nanometers,

r
= Bt1000

By transposition,
we may solve for either t or B,

t
=

r

B1000

B
=

r

t1000

Normally, we don’t
have to calculate thickness; we just use our eyepiece micrometer for the
task. Likewise, we don’t normally have to calculate retardation, because
we can compare the interference color we see in the microscope with the
interference colors on the Michel-Lévy chart, and read it off. It’s the
birefringence that we are after; that’s the identifying characteristic.
Take the common mineral quartz, for example; its thickness will vary from
grain to grain, and so its interference color will vary from one thickness
to another; but the birefringence (the numerical difference between the
principal refractive indices) is a constant; 0.009 in the case of quartz.
There it is…..get the birefringence, and you may not need any other information.
The Michel-Lévy chart is a rapid way of determining birefringence from
the particle thickness and the maximum interference color it is displaying.
The chart is a graphical solution to the above equations.

click image to view large PDF version (581K)

figure 4

Let’s look a little
more closely at the Zeiss rendition of the Michel-Lévy chart – Figure
4. Notice that the thickness, t, increases along the ordinate on
the left side of the chart from 0 µm to 50 µm, and is numbered at 10 µm
intervals. Along the bottom of the chart, left to right, the path difference,
or retardation, r, increases from 0 nm to over 1744 nm, and is marked
at specific values of retardation, 0, 40, 97, 158, 200, 218, 234 …… 1744nm
(this chart uses the older designation millimicrons, mµ, for nm). The
names of the interference colors are also given. First-order red falls
at about 550 nm; second-order red at two times 550, or 1,100 nm; third-order
red at three times 550, or 1,650 nm, etc. The birefringence (n2
- n1) is plotted along the top and right side of the
chart; it starts at zero at the upper-left corner, and proceeds to the
right 0.001, 0.002, 0.003, 0.004 …… to 0.036 at the upper-right corner,
and then proceeds downward, graduated differently, 0.040, 0.045, 0.050,
0.055 …... down to 0.180, and beyond.

Notice the family
of diagonal lines originating at the lower left and going out to each
value of birefringence. The diagonal lines represent the birefringence
values, because t
= r/B is the equation for a straight line through the origin of the coordinates
with a slope of tan θ = 1/B. Each line is assigned an angle θ,
and thereby a special value B. The names of many substances appear opposite
their characteristic maximum birefringence value on many versions of the
chart. On this Zeiss chart (Figure 4), minerals are listed by
their identifying birefringence. We mentioned quartz earlier; its name
is listed at 0.009; spodume is at 0.020; olivine is at 0.036; calcite
is near the bottom-right of the chart, near 0.180 – a high birefringence;
a large numerical difference between its refractive indices (omega = 1.6584;
epsilon = 1.4864; 1.6584 – 1.4864 = 0.1720).

KNOW
TWO, FIND THE OTHER

Since the Michel-Lévy
chart shows the interrelationships between thickness, birefringence, and
interference color, the microscopists can, as suggested earlier, determine
any one from the chart if they know the other two. Several examples will
illustrate this:

Example 1:
Suppose a cylindrical synthetic fiber 15 µm in diameter shows a maximum
interference color corresponding to about 900 nm. This is determined
by orienting the length of the fiber at 45 degrees to the vibration directions
of the crossed polars, and comparing the color running down the center
of the fiber to the colors in the chart. The order is found by noting
the number of reds between the center and either edge of the fiber. One
must be very careful here because the colors are very, very close together
at the edge of a cylindrical fiber. It is often better to count orders
on a taper-cut end of a fiber. In the present example, we pass through
only first-order red, indicating the yellow at the center is second order.

To determine the
birefringence, we look for 900 nm on the abscissa and move vertically
until we reach a horizontal line corresponding to a thickness of 15 µm
on the ordinate. There will be a diagonal line where the two lines intersect.
We now follow the diagonal line to the upper right to read the birefringence,
0.060, at the top of the chart. Looking up this value in a birefringence
table for synthetic fibers, we learn that a cylindrical fiber having this
birefringence is nylon. We could also have calculated birefringence from
(n2 – n1) = r/1000t:

(n2
– n1) = r/1000(t) = 900/1000(15) = 0.06

Remember, the 1000
in the equation is a units conversion factor between nanometers (retardation)
and micrometers (thickness).

Note that the thickness
of a substance, such as a crystal or fiber, must be measured along the
same direction the retardation is measured. This works well for cylindrical
fibers, as in the example above, since the measured diameter is also the
thickness. The proper thickness for non-cylindrical specimens often can
be measured by “crystal-rolling.” To do this, the specimen must be mounted
between slide and cover glass in a viscous liquid such as Aroclor 1260
(Monsanto), Karo Corn Syrup, or any mounting medium having the consistency
of molasses. Sliding the cover glass with a dissecting needle rolls the
crystal 90 degrees to a position where the thickness can be measured directly
with a calibrated eyepiece micrometer.

Some non-cylindrical
fibers and polymer films can be cut with a razor at a precise 45 degree
angle and the thickness measured as the horizontal projection of the cut.
Polymer films can be measured directly, using a thickness gauge before
mounting.

For substances having
oblique extinction, the 45 degree position to the polarizer vibration
direction needed before determining the interference color is quickly
found with many polarized-light microscopes through the use of the “45
degree click-stop lever.” The stage is rotated until the specimen goes
to extinction, then the 45 degree click-stop lever is engaged, and the
stage is again rotated until it clicks into a stop position 45 degrees
away.

Example 2:
Suppose now, we wish to predict what interference colors would be observed
on a sieved sample of the mineral wollastonite, randomly oriented in a
viscous medium in which the maximum vertical dimension is 40 µm. The
known birefringence of wollastonite from analytical tables is 0.014.
On the chart, we look along the top until we come to 0.014, where we find
a diagonal line. We follow this line down to the lower left until it
intersects a line corresponding to a thickness of 40 µm. Reading downward
at the point where these lines cross, we find the maximum interference
color to be slightly more purple than first-order red. Thus, wollastonite
particles 0 to 40 µm thick will show first-order interference colors of
black, gray, white, yellow, orange, red and purplish red depending on
their thickness and orientation.

Again, one could
solve this problem to find the maximum interference color with the equation:

R
= 1000 (0 to 40) X 0.014 = 0 to 560 nm

Example 3:
Finally, suppose we have a rock thin-section containing the mineral augite
(birefringence 0.024) showing an optic normal interference figure and
a first-order red interference color (550 nm). It is desired to know
the thickness of the section. At the top of the chart, we find the birefringence
0.024 and follow the diagonal line until it intersects the 550 nm line
on the abscissa. From the coordinates, we go directly left to the thickness
on the ordinate and find 23 µm. Once again, we can find the solution
from the equation:

t
= 550/1000 (0.024) = ~23 µm

Other orientations
of augite would show lower order interference colors.

Thus, the Michel-Lévy
Birefringence Chart enables the microscopist to rapidly determine thickness,
birefringence, or retardation, knowing the other two quantities.

The original chart
was used for aiding in the identification of minerals that make up rocks.
The standard rock thin-section is 30 µm thick; thus, the chart, which
is applicable up to 50 µm, is more than adequate for rock thin-section
work. A rock section is determined to be 30 µm thick by grinding it down
while periodically stopping to observe the decreasing interference colors
of ubiquitous quartz (birefringence 0.009) until they show a maximum interference
color of pale straw-yellow. Many industrial specimens, soil minerals,
etc. are larger (thicker) than 50 µm. For these specimens, one can simply
extrapolate. For example, if a comminuted quartz grain shows second-order
blue colors, how thick is it? Extend the 0.009 diagonal birefringence
line beyond the top of the chart until it intersects a vertical line at,
say 650 nm (blue), also extended beyond the top of the chart. At the
point where the two lines intersect, come straight left to an extended
thickness line, and estimate the thickness to be about 70 µm (it calculates
to 72.2 µm).

A problem arises
in using the Michel-Lévy interference color chart when it comes to determining
the exact order of a particular color, especially with thicker specimens
of high birefringence. This problem is overcome through the use of “accessory
plates” or “compensators.” These accessory retardation plates are interposed
in the light path, usually in a slot between the microscope objective
and the analyzer, to help determine the order of a particular color. These
plates may be made of mica, selenite, quartz, or calcite. They help to
determine the order by adding or subtracting either fixed or variable
amounts of known retardation to the retardation shown by the specimen.
Those with a fixed amount of known retardation, such as the quarter-wave
plate (~137 nm) and first-order red plate (~550 nm), help determine the
first order and, to some extent, the second order. A wedge made from
quartz may determine three, four, five, six, or even up to seven orders.
When we come to fractions of a wavelength, say λ/30, or many wavelengths,
say 120 λ, however, we need rotary accessory plates, such as the
Ehringhaus and Brace-Köhler.

For the Olympus BX-51
polarizing microscope I am using for the AtlasofMicroscopicParticles, I have two rotary Berek compensators; one measures very
accurately within the range 0-3 orders, and the other measures within
the range 0-20 orders. The Senarmont compensator measures very accurately
retardations within 1
order, using monochromatic light and the rotating graduated analyzer.
The entire subject of compensation is deserving of treatment, and an article
in this series on Senarmont compensation will appear shortly.